MathDB

concentric circumcircles wanted, starting with isosceles trapezoid

Source: Iranian Geometry Olympiad 2021 IGO Elementary p4

1/25/2022

In isosceles trapezoid

ABCD

(

AB \parallel CD

) points

E

and

F

lie on the segment

CD

in such a way that

D, E, F

and

C

are in that order and

DE = CF

. Let

X

and

Y

be the reflection of

E

and

C

with respect to

AD

and

AF

. Prove that circumcircles of triangles

ADF

and

BXY

are concentric.
Proposed by Iman Maghsoudi - Iran

geometryreflectiontrapezoidconcentric circles

Igo 2021 intermediate p4

Source: Intermediate p4

12/30/2021

Let

ABC

be a scalene acute-angled triangle with its incenter

I

and circumcircle

\Gamma

. Line

AI

intersects

\Gamma

for the second time at

M

. Let

N

be the midpoint of

BC

and

T

be the point on

\Gamma

such that

IN \perp MT

. Finally, let

P

and

Q

be the intersection points of

TB

and

TC

, respectively, with the line perpendicular to

AI

at

I

. Show that

PB = CQ

. Proposed by Patrik Bak - Slovakia

geometry

any circle passing through two points contains at least 673 of others

Source: IGO 2021 Advanced P4

1/2/2022

2021

points on the plane in the convex position, no three collinear and no four concyclic, are given. Prove that there exist two of them such that every circle passing through these two points contains at least

673

of the other points in its interior. (A finite set of points on the plane are in convex position if the points are the vertices of a convex polygon.)

geometryIGO

4

Problems(3)